Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm
whose coefficient is 1. Where possible, evaluate logarithmic expressions.
-$$\frac { 1 } { 4 } \left( \log _ { 8 } x + \log _ { 8 } y \right) - 6 \log 8 ( x + 1 )$$
A) $\log _ { 8 } \frac { \sqrt [ 4 ] { x y } } { ( x + 1 ) ^ { 6 } }$
B) $\log 8 \frac { \sqrt [ 4 ] { x + y } } { ( x + 1 ) ^ { 6 } }$
C) $\log 8 \frac { \sqrt [ 4 ] { x y } } { 6 ( x + 1 ) }$
D) $\log _ { 8 } \frac { \sqrt [ 4 ] { x } + \sqrt [ 4 ] { y } } { ( x + 1 ) ^ { 6 } }$

Question

Quizplus · Accepted Answer

Using properties of logarithms, the expression can be condensed as follows: The sum inside the logarithm can be converted to a product using the property $\log_b(m) + \log_b(n) = \log_b(mn)$, and the coefficient $\frac{1}{4}$ indicates a fourth root. The subtraction of $6\log_8(x+1)$ translates to division inside the logarithm and raising $(x+1)$ to the 6th power due to the property $a\log_b(m) = \log_b(m^a)$. Therefore, the correct answer is $\log_8 \frac{\sqrt[4]{xy}}{(x+1)^6}$.

Use Properties of Logarithms to Condense the Logarithmic Expression 14(log⁡8x+log⁡8y)−6log⁡8(x+1)\frac { 1 } { 4 } \left( \log _ { 8 } x + \log _ { 8 } y \right) - 6 \log 8 ( x + 1 )41​(log8​x+log8​y)−6log8(x+1)

Use Properties of Logarithms to Condense the Logarithmic Expression $\frac { 1 } { 4 } \left( \log _ { 8 } x + \log _ { 8 } y \right) - 6 \log 8 ( x + 1 )$